Question

Sketch a parabola thats reflected over the x-axis, shifted 11 units up and 6 units right

Answer 1

The parabola reflected over the x-axis, shifted 11 units up, and 6 units right can be represented by the equation y = a(x - 6)² + 11, where
`\(a\)` is the coefficient determining the direction of the reflection.

Step-by-step explanation:

To create a parabola reflected over the x-axis, we use the general form y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. In this case, the vertex is shifted 6 units to the right and 11 units up, so h = 6 and k = 11. Therefore, our equation becomes y = a(x - 6)² + 11.

The reflection over the x-axis is achieved by changing the sign of the
`a` coefficient. If the original parabola opens upward, the reflected parabola will open downward, and vice versa. Since the direction of reflection is not specified, we assume a reflection over the x-axis. Therefore, the final equation is
`\(y = a(x - 6)^2 + 11\)`, where
`\(a\)` determines the steepness and direction of the parabola.

In summary, the parabola reflected over the x-axis, shifted 11 units up, and 6 units right is represented by y = a(x - 6)² + 11, where
`\(a\)` determines the direction of the reflection. This equation provides a clear and concise expression for the given transformation of the original parabola.

Answer 2

The parabola, when reflected over the x-axis, shifted 11 units up, and 6 units right, maintains its typical U-shape with the vertex moved to the new coordinates.

Step-by-step explanation:

Reflecting a parabola over the x-axis flips its orientation, making the upward curve downward. Shifting it 11 units up means the entire parabola is elevated vertically, while a 6-unit right shift moves the parabola horizontally to the right. These transformations alter the position of the vertex, and in this case, it's shifted 6 units to the right and 11 units up.

The general form of a parabola, y = ax² + bx + c, remains the same, but the coefficients and constants are affected by the transformations.

The reflection is achieved by negating the coefficient of the x² term, resulting in a mirrored image across the x-axis. The upward shift is incorporated by adding 11 to the constant term (c), and the right shift is implemented by adjusting the x term. These transformations are part of the broader study of conic sections in mathematics, specifically dealing with parabolas and their properties.